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Chapter 7
Planet formation
The observations indicate that planets form in circumstellar disks around young stars.
Therefore, planet formation is strongly linked to the star formation process which we
describe in the introductory section of this chapter. In addition, we discuss basic properties
of interstellar and circumstellar material and then we treat circumstellar disks and the
planet formation within disks.
7.1
Star formation
7.1.1
Components in the interstellar medium
Molecular clouds are a basic prerequisite for star formation because gravitational collapse
of diffuse gas to stars occurs only in dense, cold clouds. Molecular clouds are located in the
mid-plane of the Milky Way disk. The different types of gas components of the interstellar
medium (ISM) are listed in Table 7.1.
Table 7.1: Components of the interstellar medium (ISM) in the Milky Way disc.
T [K]
N (H)[cm−3 ] gas type
main particles
1.
2.
10 − 100
100 − 1000
103 − 106
≈ 1 − 10
molecular clouds
diffuse atomic gas
3.
4.
≈ 10000
≈ 10000
10 − 104
≈ 0.1
H ii-regions
diffuse, photo-ionized
gas
diffuse, collisionally ionized gas
H2 , dust, CO, ...
H0 , dust, C+ , e− , N0 , O0 ,
...
H+ , e− , dust, X+i , ...
H+ , e− , dust, X+i , ...
5.
6
>
∼ 10
≈ 10−3
H+ , e− , X+i , ...
Pressure equilibrium. The diffuse cold (100 K), warm (10’000 K) and hot components
(106 K) are in rough pressure equilibrium
pISM = nkT
p
= nT ≈ 1000 [K/cm3 ] .
k
1
(7.1)
Cold, warm and hot phases. The interstellar gas exists predominantly in three temperature regimes:
– cold gas with T < 1000 K (typical is 10-100 K),
– warm gas with T ≈ 100 000 K,
– hot gas with T > 106 K.
The existence of these three phases is due to the cooling function Λ(T ) for astrophysical
gas. Gas radiates due to the conversion of thermal energy via particle collisions into
radiation energy which escapes from the region. The cooling is proportional to the particle
density squared, because two particles are required for a collision. The cooling rate is
C = n2 Λ(T )
which is given e.g. in [erg/cm3 s] for the energy lost per unit volume and time. Most
efficient cooling mechanisms are:
– emission of CO molecular lines for cold molecular clouds and far-IR atomic finestructure lines from C II and O I for cold atomic gas,
– emission of nebular lines from ionized atoms, like O II, N II, S II, O III, C IV and
others for warm gas,
– bremsstrahlung for the hot gas.
Figure 7.1: Specific cooling function and the predominant cooling and heating processes.
The heating of the gas can be due to very different processes. For certain temperature
and density regimes there are the following predominant heating processes:
– collisions by cosmic rays, photodissociations of molecules, and gas turbulence for
cold, dense gas, and photo-ionization for cold, diffuse (UV-transparent) gas,
– photo-ionization by stellar UV radiation for warm gas,
– adiabatic shocks from supersonic gas motions for hot gas.
Important heating processes per volume element are proportional to the particle density
∝ nH. H is the heating function which depends on the gas temperature and density and
the incident energy in the form of radiation or gas motions from outside the gas region.
For a rough pressure and temperature equilibrium we can write:
nH = n2 Λ(T )
and p = nkT
or with n = H/Λ(T )
Λ(T )
kH
=
.
(7.2)
T
p
Temperature equilibria are reached for a broad range of heating conditions around ∼
10 − 1000 K for cold gas, ∼ 100 000 − 1000 000 K for warm gas and > 106 K for hot gas,
because for these temperature regions the specific cooling rate Λ(T )/T has a positive
gradient (see Fig. 7.1).
A positive gradient is required to maintain equilibrium, because if the gas temperature
is slightly perturbed upwards, then the cooling becomes more efficient than the heating,
so that the gas temperature is returned to the equilibrium point. On the other hand,
the cooling is less efficient than heating for slightly lower temperatures, so that the gas
temperature remains close to the initial state.
If the gradient of Λ(T )/T is negative, then the cooling is less efficient than heating for a
slightly enhanced temperature and the gas starts to heat up. Conversely, if the temperature
is slightly perturbed downwards, then the cooling becomes more efficient than heating and
the gas starts to cool down.
Distribution of the different gas components. The diffuse atomic gas and the
collisionally ionized hot gas (components 2 and 5 in Table 7.1) fill most of the space in
the Milky Way disk, while the molecular gas and the cool atomic gas (components 1 and
2) make up about 90 % of the baryonic mass.
The molecular clouds and H II regions (components 2 and 3) are over-dense regions in
the interstellar medium. Molecular clouds are the locations where star formation occurs.
They are localized in the mid-plane of the galactic disk, preferentially but not exclusively
in the spiral arms. High mass stars > 10 M , which are newly formed in molecular clouds,
are very hot and they emit enough UV radiation to ionize their parent cloud. In this way,
the bright H II regions, like the Orion nebula, are formed. In external galaxies, the H II
regions often trace nicely the star forming regions located along the spiral arms.
7.1.2
Molecular clouds.
Molecular clouds are overdense regions in the Milky way disk made of molecular H2 , CO
and dust predominantly. Because they are dense, their dust and gas is self-shielding the
cloud from stellar optical and UV-light from the outside. Because of this, the molecular
clouds can not be seen in the visual except for the fact that they obscure the object
behind the cloud. The best way to see molecular clouds are CO line observations at
λ = 2.6 mm in the radio range (see Slide 7.1). The dark irregular bands of absorption in
the Milky Way are due to these absorbing clouds. The following types of molecular clouds
are distinguished:
– Bok globules are small, isolated, gravitational bound molecular clouds of <
∼ 100 M
in which at most a few stars are born,
– molecular clouds have masses of 103 − 104 M distributed in irregular structures
with dimensions of ≈ 10 pc consisting of clumps, filaments bubbles and containing
usually hundreds of new-born stars,
– giant molecular clouds are just larger than normal molecular clouds with a total
mass in the range 105 − 107 M , dimensions up to 100 pc, and thousands of young
stars.
Molecular clouds in the solar neighborhood. The sun resides in a hot (106 K), low
density bubble with a diameter of ≈ 50 pc. The nearest star forming clouds are located
at about 140 pc and because of their proximity they are important regions for detailed
investigations of the star and planet formation process. Well studied regions are:
– The Taurus molecular cloud, at a distance of about 140 pc, is a large, about
30 pc wide, loose association of many molecular cores with a total mass of about
≈ 104 M and several hundred young stars. Because of its proximity there are many
well known prototype objects, like T Tauri or AB Aurigae in this star forming region.
– The ρ Oph cloud at a distance of 130 pc is composed of denser gas than Taurus
with a main core and several additional smaller clouds and about 500 young stars
with an average age of about 0.2 Myr. The total gas mass is about ≈ 104 M .
– the Orion molecular cloud complex has a distance of about 400 pc and a diameter
of 30 pc. Orion is the nearest high mass star forming region with in total about 10’000
young stars with an age less than 15 Myr. The Orion molecular cloud complex
includes the Orion nebula, or M42 (H II region), reflection nebulae, dark nebulae
(Horsehead nebula), and OB associations mainly located in the Belt and Sword of
the Orion constellation. The Orion nebula is ionized by the brightest star in the
Trapezium cluster.
7.1.3
Elements of star formation
Stars form in dense, molecular clouds. If regions in clouds become dense enough, then they
may collapse under their own gravitational attraction and form a dense concentration of
gas, which evolves into a star. The star formation process is very complex involving many
different physical phenomena like the interaction of gas with radiation, hydrodynamics,
magnetic fields, gas chemistry, dust grain evolution, gravity and more.
Key parameters of the gas must be strongly changed for a transition from a cloud to a
star:
– the density of a cloud must be enhanced from ∼ 10−20 g cm−3 to about 1 g cm−3 in
a star,
– the specific angular momentum (per unit mass) of the gas must be lowered from
∼ 1022 cm2 s−1 to about ∼ 1020 cm2 s−1 for a binary system or ∼ 1017 cm2 s−1 for a
single star with a planetary system,
– and the magnetic energy per unit mass must be lowered from about ∼ 1011 erg g−1
to about ∼ 10 erg g−1 .
Thus, star formation means that the gas is strongly compressed by self-gravity, that it must
loose essentially all of its angular momentum by fragmentation and magnetic breaking,
and it must be strongly de-magnetized by processes like ambipolar diffusion.
Gravitational equilibrium and Jeans mass. Because molecular clouds are longlived, they must be not only in pressure-temperature equilibrium, but also in hydrostatic
equilibrium. The virial theorem is valid for systems in a gravitational equilibrium.
2 Ekin + Epot = 0
(7.3)
If we consider a homogeneous (constant density), spherical and isothermal cloud then the
kinetic (or thermal) energy Ekin = Etherm = 3kT M/2µ. This yields for the virial theorem:
2·
3 k
3 G M2
TM−
=0
2µ
5 R
(7.4)
This can be rearranged into kT /µ = GM/5R. Let us substitute the radius using the mean
density of a homogeneous sphere (ρ = 3M/4π R3 ) and solve for M and ρ. This provides
an estimate for the equilibrium density or equilibrium mass for a given gas temperature
T . These quantities are called Jeans-mass:
MJ =
375 1/2 k
3/2 1
T
.
4π
Gµ
ρ1/2
and Jeans-density:
ρJ =
3 1
375 k
T
4π G µ
M2
Example: The Jeans-density for M = M , T = 10 K and µ = 2.7 is ρJ ≈ 7 · 10−19 g cm−3
equivalent to a particle density (H2 ) of 2 · 105 cm−3 .
For a fixed cloud temperature and density, the Jeans mass gives the mass needed for the
cloud to be in gravitational equilibrium. The Jeans mass is smaller for cold, high density
clouds. Similarly, for a given cloud mass and temperature, the Jeans density describes the
density required to be in a gravitational equilibrium. The density can be rather low for
high mass, cool clouds.
The Jeans density and Jeans mass are parameters for an interstellar cloud in a gravitational
equilibrium. However, it is not clear whether this equilibrium state is stable or whether
even a small perturbation can yield a collapse to a star or an expansion and diffusion of
the cloud.
In a closed box model the cloud remains in a hydrostatic equilibrium. If the cloud
is slightly compressed then the liberated potential (or gravitational) energy is converted
into thermal energy, which enhances the gas pressure and the system goes back into the
equilibrium state.
Contraction by radiation. A contraction is possible if energy is radiated away. If
contraction occurs then potential energy is converted into kinetic energy ∆Ekin = −∆Epot
and if part of this thermal (or kinetic) energy is radiated away then the system can find
a more compact quasi-equilibrium configuration. According to the virial theorem, half of
the liberated potential energy must be radiated away, while the other half is converted
into thermal energy
1
Lcloud = − ∆Epot
2
and
1
∆Ekin = − ∆Epot .
2
(7.5)
The contraction speed depends on the radiation or cooling time-scale:
3
1
3 kT
τcooling ≈ nkT 2 =
.
2
n Λ
2 nΛ
The cooling time scale becomes shorter during the collapse because the particle density
increases steadily.
– the contraction is rapid in the optically thin case, because then the radiation can
escape from the entire cloud volume,
– the contraction is slow if the cloud is optically thick, because the radiation can
only escape from the surface.
The virial theorem requires that the cloud temperature raises during contraction, if no
radiation is emitted. Thus, contracting clouds heat up. But, because warmer gas emits
more efficiently (as long as it is below T . 1000 K) for higher temperatures (see Fig. 7.1),
the luminosity and therefore the loss of radiation energy of the contracting object becomes
higher until the fast contraction changes into a slow quasi-static contraction when the cloud
becomes optically thick.
Stabilization mechanisms must exist for self-gravitating clouds because otherwise all
existing clouds would collapse in a short timescale. Mechanisms which can stabilize a
cloud against collapse are:
– cloud heating processes, like radiation from external stars, cosmic rays, magnetohydrodynamic turbulence and sound waves, which enhance the gas temperature and
the gas pressure so that the cloud expands;
– angular momentum conservation may inhibit collapse because of enhanced centrifugal effect for more compact and therefore more rapidly rotating clouds;
– magnetic fields, if there are ions in the molecular cloud, so that the magnetic
fields are frozen into the plasma and contraction enhances the magnetic pressure
2
.
like pmagn ∝ B02 /rcloud
Star formation is complicated because so many different processes play a role and from
observations it is often hard to get detailed information about the cloud geometry, heating
processes, specific angular momentum, and magnetic properties of a gas. A few important
aspects of star formation follow from the stabilizing processes discussed above.
Star formation feedback is the influence of new-born stars on their environment. Young
stars have strong outflows and emit energetic radiation which both can heat the surrounding cloud and stop the star formation process. On the other hand, this heating produces
over-pressurized bubbles, like the Orion nebula, which expand and which may compress
the adjacent gas and trigger the collapse of a cloud. Depending on the details positive or
negative feedback occurs and there is strong observational evidence that both mechanisms
occur. However, many aspects of the star formation feedback are still unclear.
Fragmentation is linked to the Jeans mass. If a cloud contracts isothermally (loss of
energy through radiation) then the density increases and the Jeans mass becomes smaller
√
like MJ ∝ 1/ ρ. Thus, a large contracting cloud can decay into smaller clouds so that
many stars are formed simultaneously in a big cloud complex. Typically there are many
low mass stars formed M < 1 M but only a few high mass stars M > 1 M .
.
Figure 7.2: Schematic illustration of the fragmentation process.
The specific angular momentum of the gas in a molecular cloud is very large when
compared to a contracted proto-stellar clouds. Therefore, the angular momentum barrier
inhibits a global contraction of a cloud. However, if sub-units can collapse into proto-stars
then the global angular momentum with respect to the entire cloud is preserved as motion
of the proto-stars around the center of gravity. The remaining specific angular momentum
of the gas with respect to the individual proto-stellar cloud unit is then much smaller. The
angular momentum barrier is a second important aspect in favor of cloud fragmentation
and the quasi-simultaneous formation of many stars from big molecular cloud.
Proto-stellar disks and binaries are a further result of the angular momentum conservation. A contracting pre-stellar cloud core needs still to get rid of angular momentum.
Angular momentum transfer via magneto-hydrodynamic processes helps to transport angular momentum away from the contracting cloud.
Another option is the formation of a binary star or a circumstellar disk. Both are
configurations which can “store” more angular momentum than a rapidly rotating star.
Figure 7.3: Schematic illustration of ambipolar diffusion.
Ambipolar diffusion can solve the magnetic field pressure problem. A contracting cloud
with charged particles also “clamps” the galactic magnetic field and will, therefore, soon
“feel” the magnetic pressure which acts against further contraction. The magnetic field
can move out of a neutral molecular clouds by the so-called ambipolar diffusion. In the end,
this leaves a compact, demagnetized, cloud core. The fact that stars form predominantly
in dense, cold, neutral clouds could be due to the lack of magnetic pressure.
7.1.4
Initial mass function
Collapsing interstellar clouds form stars in the mass range from 0.1 to 100 M . The initial
mass function (IMF) describes the mass distribution for the formed stars. According to
the classical work of Salpeter (1955), this distribution can be described for stars of about
solar mass and above with a potential law of the form:
dNS
∝ M −2.35
dM
for M > M .
(7.6)
This relation is often given as a logarithmic power law of the form
dNS
∝ M −1.35
d log M
because
dNS
dNS d log M
1 dNS
=
=
.
dM
d log M dM
M d log M
This is equivalent to a linear fit with slope −1.35 in log M -log NS diagram (Figure 7.4).
This law indicates that the number of newly formed stars with a mass between 1 and 2
M is about 20 times larger than the stars with masses between 10 and 20 M . If we
consider the gas mass of the molecular cloud, then about twice as much gas ends up in
stars between 1 and 2 M when compared to stars with masses between 10 and 20 M .
The initial mass function seems to be valid for many regions in the Universe, for the star
formation in small molecular clouds, larger cloud complexes, and the largest star forming
regions in the local Universe. Up to now no star forming regions have been found for
which the Salpeter IMF is a bad description.
For low mass stars, the mass distribution shows a turn over. Since M-stars M < 0.5 M
have a main-sequence life time which is longer than the age of the universe we can just
use as first approximation the frequency of stars with different spectral types as rough
description for the IMF of low mass stars (see also Table 7.2). This distribution shows a
maximum in the range of M3V to M5V stars. Thus, the mass distribution has a turn-over
at a mass of about 0.4 M followed by a rapid drop-off towards substellar objects. The
minimum is around 0.01 M or 10 MJ at the lower end of the brown dwarf regime. In the
planet regime, the frequency of object starts to raise again strongly towards lower masses
at least down to ME as demonstrated by the planet transit statistics from the Kepler
satellite.
Figure 7.4: Schematic illustration of the initial mass function for stellar and substellar
objects.
Table 7.2: Statistics for the stellar systems within the solar neighborhood up to a distance
of 10 pc.
object type
numbers
MV (spec. type)
stars (total)
O and B stars
A V stars
F V stars
G V stars
K V stars
M V stars
G and K III giants
white dwarf stars
342
0
4
6
20
44
248
0
20
–4.0 (B0 V)
+0.6 (A0 V)
+2.7 (F0 V)
+4.4 (G0 V)
+5.9 (K0 V)
+8.8 (M0 V); +12.3 (M5 V)
+0.7 (K0 III)
single stars
binary systems
multiple systems (3+)
185
55
12
brown dwarfs
15
planets
planetary systems
19
11
Considering the complex physics involved in the star formation process, it is surprising
that the initial mass function is such an universal law that seems to be valid everywhere
in the Universe. There must be one essential process that dominates the outcome of the
stellar mass distribution. This could be the fragmentation process. Further it seems to
be clear that there are different regimes of formation between stars and planets. The low
frequency of substellar object in the mass range 0.01 - 0.1 M indicates that such objects
are not easily formed via the normal star forming process, perhaps because the formation
of small fragments or their survival in molecular clouds is rather unlikely. On the other
hand, gas giant planets are very common but seem to form predominantly around stars.
This indicates that there exists a bimodal formation mechanism of hydrostatic astronomical objects.
– stars are formed by the collapse and fragmentation of clouds,
– planets are the result of a formation process in circumstellar disks.
7.1.5
Types of proto-stars
There are different phases in the star formation process, from a collapsing cloud, to a
pre-stellar core, to a proto-star, and a pre-main sequence star. Some of these phases have
specific observational characteristics in the spectral energy distribution (SED). The SED
show the signatures of the following components:
– the Planck-spectrum of the main energy source, with its characteristic peak flux
wavelength for the temperature of the object,
– an infrared excess, if optical to near-IR radiation is absorbed by the circumstellar
material and re-radiated at longer wavelength,
– an UV-visual excess because of energetic processes due to gas accretion onto the
star,
– emission lines if the energetic processes are strong enough to dissociate and ionize
gas.
Figure 7.5: Schematic illustration of the spectral energy distribution for the different types
of young stellar objects.
According to the presence and characteristics of these features different types of young
stellar objects are distinguished:
– Class 0: The SED peaks in the far-IR or sub-mm part of the spectrum near 100 µm
(30 K), with no flux in the near-IR. These are the dense, pre-stellar cloud cores.
– Class I: They have a flat or rising SED from about 1 µm towards longer wavelengths
indicating that a hot source (≈ 1000 K) is still embedded in a cloud, so that most
radiation from the young stellar object is absorbed and re-radiated as IR-emission
by the circumstellar dust.
– Class II: They have falling SED into the mid-IR and the underlying objects have
the characteristics of so-called classical T Tauri stars or Herbig Ae/Be stars. They
exhibit strong emission lines and often a strong UV excess from the accretion process.
These are the systems with extended circumstellar disks that are strongly irradiated
by the central proto-star.
– Class III: These are pre-main-sequence stars with little or no excess in the IR, but
still showing some weak emission lines due to gas accretion. One of the subgroups
of this class are the weak-lined T Tauri stars.
Class II and Class III objects can be placed into the Hertzsprung-Russell diagram if the
temperature and luminosity are corrected for the contribution from the accretion processes.
Compared to normal, main-sequence stars the Class II and Class III objects are located
above the main sequence. These object evolve then “down” to the main-sequence.
The pre-main-sequence time scale, which describes the quasi-static contraction of a
young star follows from the Virial theorem. The Virial theorem requires that half of the
potential energy gained by the gravitational contraction is radiated away as described by
τKH ≈
Epot
G M2
≈
.
L
RL
(7.7)
This time-scale is also called the Kelvin-Helmholtz timescale. For solar parameters there
is τKH ≈ 30 Myr. Pre-main sequence stars start as relatively large ≈ 3 R and luminous
objects ≈ 10 L with correspondingly shorter time-scales.
7.2
Circumstellar disks
Before 1980 there existed only indirect evidence about the presence of circumstellar disk
around young stars. In the 1980’s collimated outflows, so-called jets, could be clearly
associated with young stars (Slide 7.5). Jets are a well known phenomenon of accretion
disks in active galactic nuclei and binaries where mass flows from one component through
an accretion disk onto the companion.
Direct imaging of accretion disk became only possible with high resolution observations
using HST. Edge-on disks could be imaged in nearby star forming regions and in Orion
one could see the dark, light absorbing silhouettes of dusty disks in front of the extended
nebular region (Slide 7.6 and 7.7). Thanks to coronagraphs on HST and ground based AO
instruments with high contrast capabilities it is now possible to take images in scattered
light for face-on proto-planetary disks. Some of the best images of such images were taken
by our group in ETH (Slide 7.8). Well resolved images of the gas in proto-planetary disks
are now also possible in the mm-wavelength range using the new ALMA interferometer.
Another type of disk was discovered with the first far-IR satellites. It was recognized
that many (10 %) A-stars exhibit an infrared excess. Young main sequence star β Pic
is the prototype of these stars and it shows a disk of dust, similar to the zodiacal dust
disk in the solar system (Slide 7.9). The dust in these disk is created by colliding solid
bodies (e.g. asteroids and meteoroids) or evaporating material from comets. Such disk are
frequent around young stars and they are signposts of the early evolution of a planetary
system.
All these data show, that disks are an important feature in the star and planet formation process.
7.2.1
Constraints on the proto-planetary disk of the early Solar System
We can use the current Solar System to infer some limits on the angular momentum and
the mass of the circumstellar disk which formed “our” planetary system.
Angular momentum. The angular momentum budget of the Solar System can be split
into the angular momentum of the Sun and the angular momentum of the planets.
The angular momentum of the Sun can be calculated as for a rotating, spherically symmetric sphere
Z
2
LS =
(~r × (~
ω × ρ(r)~r))dV = k M R
ω = 3 · 1048 g cm2 s−1
V
where k = 0.1 accounts for the radial density distribution ρ(r) of the sun (k = 2/5 for
homogeneous sphere), and ω = 2.9 · 10−6 s−1 = 2π/25 days is the angular velocity of the
solar rotation.
This can be compared to the angular momentum of Jupiter, which is
p
1
LJ = aJ MJ vJ = MJ GM aJ = 2 · 1050 g cm2 s−1 .
2
The total angular momentum of Jupiter is almost 100 times larger than for the Sun. For
the specific angular momentum the contrast is even more dramatic with the material in
the planets having about 105 times more specific angular momentum than the gas in the
Sun. Thus, there must exist an efficient process which can segregate angular momentum
during the formation of the Solar System.
Mass distribution. The Sun contains more than 99 % of the mass of the Solar System.
We can deduce how much mass was present in the circumstellar disk for the formation of
the planets. A rough budget for the gas mass and the mass of heavy elements is given in
Table 7.3. This budget provides:
– the mass of the heavy elements present now in the planetary system;
– an estimate on the total gas mass available in the pre-solar nebula for planet formation, if the dust to gas ratio was md /mg ≈ 0.01;
– a comparison with the gas mass and heavy element mass of the sun.
Table 7.3: “Minimum mass limit” for the circumstellar disk required for the formation of
the planetary system.
total mass budget
heavy element (h.e.) budget
1033
mass of the Sun
mass of giant planets
mass of terr. bodies
2·
g
30
3 · 10 g
3 · 1028 g
mass required for planets
for mg /md = 100
3.3 · 1031 g
1 % h.e. in the Sun
10 % h.e. in giant planets
100 % h.e. in terr. bodies
h.e. in all planets
2 · 1031 g
3 · 1029 g
3 · 1028 g
3.3 · 1029 g
As Table 7.3 indicates, the heavy elements in the solar system are predominantly > 98 %
located in the Sun. The required gas and dust mass for the formation of the planets was
about 1 % of the mass of the Sun. This quantity if often called the “minimum mass solar
nebula”. Thus, the required reservoir of mass for the planet formation was small when
compared to the mass required for the Sun. A disk mass of about 1 % of the mass of
the central star is in rough agreement with the mass estimates from observations of disks
around young stars.
For our model we distribute the 0.01 M minimum mass for the solar proto-planetary disk
in rotationally symmetric uniform annuli. In the radial direction, in terms of mass surface
density, we adopt the distribution of the heavy element mass as the current Solar System
multiplied by the gas to dust ratio mg /md . This yields a disk with a surface density
distribution of gas according to
r −3/2
Σ = 2 · 103
g cm−2
AU
extending roughly from the orbit of Venus to the orbit of Neptune. At a separation of
1 AU this disk would consist in vertical direction of about 20 g of dust per cm2 .
Model disk description. For an initial discussion the description of circumstellar disks
can be simplified with several reasonable assumptions:
– the disk is considered to be rotationally symmetric and symmetric with respect to
the disk plane and can be described in an r-z-coordinate system,
– the vertical structure is given by the hydrostatic equilibrium
dP
= −ρ(r, z)gz
dz
(7.8)
– the gas in the disk is assumed to move in quasi-Keplerian
orbits and the specific
p
3
angular momentum of the disk gas is (using ω = GMS /r )
p
`(r) = r2 ω = GMS r .
– for thin, low mass disks, the gz gravity vector component (perpendicular to the
plane of the disk) is dominated by the central star, and is proportional to tan θ =
z/r ≈ sin θ.
GMS
GMS
gz =
sin θ =
z = ω2z
2
r
r3
– if the vertical density structure for the thin disk is isothermal, then P ∝ ρkT /µ =
ρc2s , where cs is the sound speed. Then, the hydrostatic equilibrium becomes:
dρ
ω2
= −ρ 2 z
dz
cs
with the solution
ρ(r, z) = ρ0 e−z
2 /2h2 (r)
,
where h(r) = cs (r)/ω(r) is the vertical scale height and ρ0 = ρ(r, z = 0).
– The surface density
Z +∞
Σ(r) =
ρ(r, z) dz
−∞
is used to describe the radial structure of the disk.
(7.9)
(7.10)